Nuclear magnetic resonance may be used to detect certain properties of a sample having atoms with nuclear magnetic moments associated with non-zero nuclear spins. In the absence of a magnetic field, the equilibrium orientations of these nuclear moments are random and the energies associated with different orientations of a nuclear moment are small. In the presence of a static magnetic field, that these nuclear magnetic moments assume certain allowed quantized orientations with respect to the static magnetic field. These quantized orientations give rise to distinct quantized energy levels known as nuclear Zeeman levels.
A transition, may be induced between two Zeeman levels by illuminating a sample with electromagnetic radiation in resonance with the transition between two Zeeman levels. The orientation of a nuclear moment can be changed by such a resonant excitation. This phenomenon is called nuclear magnetic resonance ("NMR"). In practical applications, the magnitude of the static magnetic field is usually within a range such that the energy separation between two adjacent nuclear Zeeman levels corresponds to the radio frequency ("rf") spectrum. Accordingly, a time-varying magnetic field for modulating the magnetization of a sample is within the rf range. Conventionally, the strength of the interaction between the applied rf field and the spin state of a sample is characterized by a Rabi frequency, which corresponds to the coupling energy.
Nuclear spins are intrinsic properties of atomic nuclei and can be used as part of noninvasive probes for analyzing many materials. Indeed, NMR spectroscopy and NMR imaging have been widely used to analyze the electronic and molecular structure, motion, and chemical composition of a sample.
Most of the existing NMR systems use two different types of configurations, magnetic induction detection and magnetic force detection. FIG. 1A shows a typical layout of a conventional magnetic induction NMR system. A sample 100 is placed in a spatially homogeneous static magnetic field B.sub.0 which may be produced by one or more permanent magnets or an electromagnet. The magnetic field B.sub.0 aligns the nuclear moments in the direction of the field B.sub.0 and thus produces an initial magnetization of the sample 100. An rf excitation coil 110 is positioned near the sample 100 to produce an rf excitation field at the sample 100. The excitation coil 110 is configured to produce the rf excitation field with a varying magnetic field perpendicular to the magnetic field B.sub.0. The rf excitation changes the initial magnetization of the sample 100 through the nuclear magnetic resonance and thereby induces an electromotive force ("emf") in a receiver coil 120 disposed near the sample 100. A current is thus generated by the Faraday induction in the receiver coil 120 and is subsequently processed as the NMR signal. In many cases, the excitation coil 110 may be the same as the receiver coil 120.
Another NMR configuration uses conventional force-detected NMR, which is in principle based on a magnetic force experienced by a magnet moment in a magnetic gradient field produced by an external magnet. The force-detected NMR can be viewed as a derivative of the well-known Stern-Gerlach force experiment in which the magnetic force on a spin magnetic dipole in a static magnetic field gradient is used to separate spin states in an atomic beam. Rabi et al., Physical Review, Vol. 53, p. 318, 1938. Similar to an induction NMR system, a rf excitation is applied to change the magnetization of a sample in the field of the external magnet in a force-detected NMR system. This in turn alters the magnetic force between the external magnet and the sample. The change in the magnetic force can be used to move a mechanical oscillator to which the sample or a sensing magnet is affixed. The position change of the oscillator is detected to produce an NMR signal by using a position sensor. The rf field modulates the magnetic force resonantly with the oscillator.
One type of conventional force-detected NMR systems is illustrated in FIG. 1B. See, Rugar et al., "Mechanical detection of magnetic resonance", Nature, vol. 360, pp. 563-566, Dec. 10, 1992; Rugar et al., "Force Detection of Nuclear Magnetic Resonance", Science, Vol. 264, pp. 1560-1563, Jun. 10, 1994; and Sidles et al., "Magnetic resonance force microscopy", Reviews of Modern Physics, Vol. 67, pp. 249-265, 1995. A sample 100 is fixed to the tip of a mechanical cantilever 130. This combination forms a mechanical oscillator. Unlike an induction NMR system, a gradient magnet 140 near the sample 100 produces a magnetic field gradient at the sample. This produces both a magnetization of the sample 100 and a magnetic force between the sample 100 and the gradient magnet. An rf excitation coil 110 located close to the sample 100 produces an rf field to modulate the magnetization of the sample 100. The position of the cantilever 130 moves as the magnetic force between the sample 100 and the gradient magnet 140 is modulated by cyclically modulating the magnetization in the sample 100 at the mechanical resonance frequency of the cantilever 130. A fiber optic sensor 150 can be used to detect this position change and produce an electrical NMR signal.
Specifically, Rugar et al. and Sidles et al. implement a cyclic adiabatic rapid passage (ARP) to invert the equilibrium longitudinal magnetization at the audio-frequency resonance of a sample-on-cantilever assembly. A ferromagnetic particle is used as the gradient magnet 140. The large magnetic field gradient is used both to provide the magnetic force, which couples longitudinal magnetization to cantilever motion, and to vary the resonance condition across the sample, which provides an imaging capability.
The force-detected NMR has been recognized to have several advantages over the induction NMR in certain applications. For example, the sensitivity of the force-detected NMR can be superior to that of the magnetic induction at small length scales and can be optimized to reach the thermal limit caused by the Brownian motion of the mechanical oscillator. Much of the interest in force-detected NMR is based on its potential for single-spin imaging which can provide a powerful alternative to induction NMR spectroscopy for determining molecular structures. This single-spin detection of the force-detected NMR if realized could be used to test single molecules and thereby the considerable work on time-consuming and costly sample preparation in the conventional NMR structural methods may be eliminated.
The conventional force-detected NMR using the sample-on-cantilever configuration (FIG. 1B) specifies a magnetic field gradient at the sample to produce the necessary magnetic force for coupling the rf energy supplied by the rf excitation coil to the oscillator. To increase the NMR signal strength, the static magnetic field needs to be large and the field gradient also needs to be large. In Rugar et. al., supra., the static magnetic field and the field gradient are typically on the order of several Tesla and several hundred Tesla per meter, respectively.
U.S. Pat. No. 5,166,615 to Sidles discloses several force-detected NMR configurations with a sample affixed onto a mechanical oscillator or vibrator. The external magnetic field applied to the sample has a gradient component. The patent states that the gradient component at the sample should be strong to enhance the spin-oscillator coupling.
Another type of the conventional force-detected NMR system places a sensing magnet on a mechanical oscillator. Similar to the aforementioned conventional force-detected NMR system, this system maintains a static magnetic field gradient at the sample to effect the magnetic force on the sensing magnet. See, for example, Zhang et al. in Applied Physics Letters, Vol. 68, pp. 2005 (1996), Smith et al. American Physical Society March Meeting Abstract, No. 19.07, 1997, and Sidles, Applied Physics Letters, Vol. 58, p. 2854 (1991).
Many practitioners in the field of NMR believe that a static magnetic field gradient at the site of a target sample is required for implementing the force detection. In particular, many have tried to improve the coupling between the nuclear spins of the sample and the mechanical oscillator by increasing the field gradient across the sample. See, for example, U.S. Pat. No. 5,166,615, supra. Hence, the magnetic field at the sample is highly inhomogeneous in conventional force-detected NMR systems. This field gradient at the sample can be limiting in several respects.
First, the variation of the magnetic field across the sample caused by the field inhomogeneity can effect spectral line broadening in the NMR signal. This is at least in part because the NMR signal corresponding to a particular transition is essentially a summation of signals of the entire sample and that transition has different frequencies at different locations in the sample. The larger the field gradient at the sample, the more the line broadening. This line broadening can significantly reduce the resolution of an NMR measurement of the conventional force-detected NMR systems. For example, some fine spectroscopic features, such as chemical shifts, spin-spin couplings, and chemical exchange effects in the NMR signal may become unresolvable due to this line broadening. Thus, field inhomogeneity at the sample is undesirable for NMR spectroscopy.
Secondly, the field inhomogeneity at the sample can reduce the coherence of the induced magnetization in the sample due to the spectral line broadening. This adverse effect can manifest itself in at least two different ways. The inventors of the present invention recognized that the magnitude of the magnetic force between the sample and the sensing magnet is proportional to the degree of coherence of the induced magnetization in the sample in addition to its linear dependence on the field gradient in prior force detection methods. Therefore, an increase in the field inhomogeneity can in fact decrease the net magnetic force. Hence, an increase in the magnetic force by increasing the field gradient is negated by the effect of the reduced coherence. Under certain conditions, increasing the field gradient at the sample to increase the detection sensitivity may actually reduce the detection sensitivity.
Another effect of the reduced coherence caused by the field inhomogeneity is that many well-developed NMR techniques may not be applicable in the conventional force-detected NMR system. This is at least in part due to the fact that many NMR spectroscopic and imaging techniques are primarily based on the coherence of the induced magnetization in the sample.
The inventors suggested that the inhomogeneous line broadening in the NMR signal may be overcome by arranging that the field gradient at the sample is turned off during the period when the spin evolution is encoded and turned on only during the subsequent detection process. See Leskokitz et al., Proceedings of the 36.sup.th Experimental NMR Conference, Boston, Mass., March 1995. This technique, however, does not solve the coherence problem during the detection period caused by the field gradient. The inhomogeneity at the sample makes contributions from different parts of the sample increasingly out of phase with respect to one another, thereby resulting in a decrease in the magnetic force.
In particular, in order to be near the optimum sensitivity for force detection on a given sample, the magnetic field strength due to the ferromagnetic particle alone may range across the sample from a maximum value comparable to that at the surface of the ferromagnet to a value of an order of magnitude or more smaller. This range, which usually is invariant to the length scale of the optimized sample-plus-particle, is of the order of 1 T or 10.sup.7 Hz in .sup.1 H Larmor frequency for the best ferromagnets. This is much greater than the maximum practical rf field strengths of about 10 mT, corresponding to .about.10.sup.5 Hz in terms of the proton Rabi frequency. The useful spectral range in various time-domain measurements is of the order of the Rabi frequency and often considerably less. Thus, the great majority of useful NMR detection methods, which are crucial to maximizing sensitivity, resolution, and information content, cannot be performed simultaneously over the entire sample volume by conventional force-detected NMR methods.
Note that this concern about loss of coherence control with inhomogeneous linewidths greater than .about.10.sup.5 Hz is quite distinct from the usual reasons for desiring homogenous magnetic fields (typically &lt;&lt;1 part per million) in high-resolution NMR. The goal of measuring fine spectroscopic features such as chemical shifts with resolution comparable to the homogeneous linewidth is the principal motivation for such highly homogeneous fields, and is neither more nor less important in the context of force-detected NMR methods.
The known reciprocal relationship between linewidth and sensitivity in directly-detected magnetic induction NMR measurements is absent in the force-detected NMR methods of the present invention. This is because modest field inhomogeneity can be made irrelevant to the signal power by utilizing a distinct detection period designed to prolong the decay of that magnetization which survives the evolution period of interest.
Consider a special case in which the longitudinal magnetization drives the mechanical oscillator at its resonance frequency .omega..sub.F as the result of cyclic adiabatic rapid passage ("ARP") repeated at 2.omega..sub.F (Rugar et al., Science, Supra.). Ideally, ARP would be used to invert the magnetization thousands of times before spin relaxation erases memory of the initial magnetization, and indeed this has been demonstrated with ordinary spectral widths of less than 1 mT and, in the context of force-detected NMR, with gradient-induced spectral widths of several mT.
Large numbers of passes in a time duration short compared to the spin-lattice relaxation time (T.sub.1) have not been demonstrated in the larger field gradients that are possible with optimized ferromagnets. But the question of whether it is practical can be answered by the following analysis. In order to prevent cancellation of forces associated with sample volume elements at different fields, the ARP should cover the entire range of Larmor frequencies in a time duration much shorter than, e.g., about one quarter of the oscillator period. This condition places a lower bound on the speed of a linear sweep, d.omega..sub.0 /dt, for a spectrum of width .DELTA. and a given oscillator resonant frequency .omega..sub.F : ##EQU1## Although the lowest practical value of .omega..sub.F is not clear, with the audio-frequency values that have been achieved to date, this will typically be a stricter lower bound than results from demanding that the sweep be completed in a time duration that is short compared to a rotating-frame spin-relaxation time, T.sub.1.sbsb..rho.. An upper bound on the sweep rate is set by the adiabatic condition, ##EQU2## where .omega..sub.1 is the Rabi frequency. Condition (2) insures that the magnetization is not irreversibly dephased during ARP. Conditions (1) and (2) place a lower bound on the Rabi frequency: ##EQU3## With .DELTA.=2.pi..times.(10 MHZ), and .omega..sub.F =2.pi..times.(500 Hz), ARP with small dissipation of spin order would require a Rabi frequency in excess of 100 kHz. Such a high Rabi frequency may be possible, but technical problems such as sample heating can render it undesirable. Systems with molecular motion comparable to the Rabi frequency can suffer greater magnetization decay during ARP than when the magnetization is longitudinal (T.sub.1.sbsb..rho. &lt;&lt;T.sub.1). Thus, it can be concluded that the desirable goal of inverting the magnetization by ARP many times in a time much shorter than T.sub.1 can be difficult or impractical for many systems and is made more difficult by the presence of the largest average field gradients that are possible for a given sample.
An alternative to ARP is a train of .pi. pulses, preferably compensated to provide accurate inversion in the presence of rf inhomogeneity and other spin interactions. In the absence of a static field gradient, this would allow a square-wave modulation of the average value of the magnetization, &lt;M.sub.z (t)&gt;, at the audio frequency of the mechanical oscillator with a modest Rabi frequency and rf duty cycle. A gradient-induced spectral width far in excess of the Rabi frequency would make such an approach ineffective. Thus this approach is not feasible in the context of previously described force-detected NMR methods, unless sensitivity is compromised by designing for far less than the largest possible force.
Homogeneity in a static magnetic field is desirable during the period in which the oscillator is being driven in proportion to &lt;M.sub.z (t)&gt; also because this would allow real-time detection of these transients. Such real-time detection of force-detected NMR may be preferred in cases where it is desirable to achieve higher throughput at the cost of reduced sensitivity. Multiple-pulse sequences may be used to achieve an effective Hamiltonian that does not commute with M.sub.z and has an audio-frequency spectrum, thus allowing spectral information to be obtained by Fourier transformation of &lt;M.sub.z (t)&gt;. The argument here parallels that in ordinary NMR where pulsed spin-locking during detection is usually the strategy that optimizes sensitivity, but requires extending the dimensionality of the time domain experiment by one. The reduction in the detection sensitivity is minimized when the NMR spectrum has multiple resolved lines and is maximal when the effective Hamiltonians provide few resolved transitions and decay is much shorter than the decay time possible in an optimized detection period.